Multiplication, weighted composition operators and their relation on disc algebra \(A(D)\) (Q2761898)

It is considered the space of all analytic in \(D=\{z\in C: |z|<1\}\) and continuous in \(\overline D\) functions, equipped with the \(\sup \)-norm. Given the operator \(T:A(D)\to A(D)\) of the form \(f\mapsto u.(f\circ\varphi)\), where \(u\in A(D)\) is a fixed function and \(\varphi \in A(D)\), \(\|\varphi \|\leq 1\) is called a weighted composition operator in \(A(D)\). The authors investigate a necessary and sufficient condition for the existence of the relation \(T=uC_{\varphi }\), where \(C_{\varphi }\) is the composition defined by \(C_{\varphi }f=f\circ \varphi \), and weighted composition operator in \(A(D)\) is defined by \(uC_{\varphi }f=u.(f\circ \varphi)\). The action of \(T^{\ast }\) on evaluation functionals is used for this purpose. Closed rangeness, compactness and relations between them are discussed as well.